A090319 Fifth column (k=4) of triangle A084938.

1, 4, 14, 52, 217, 1040, 5768, 36992, 272584, 2285184, 21550656, 226071744, 2611146384, 32911082496, 449243785728, 6598780563456, 103734755882496, 1737181702840320, 30866291090657280, 579859321408266240

Offset: 0

Author

Philippe Deléham, Feb 05 2004

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..150

Crossrefs

Cf. A084938.

Columns, for k = 0, 1, 2, 3 : A000007, A000142, A003149, A090595.

Programs

GAP

B:=Binomial;; List([0..20], n-> Sum([0..n], k-> Sum([0..k], m-> Sum([0..m], j-> Factorial(n)/(B(n,k)*B(k,m)*B(m,j)) )))); # G. C. Greubel, Dec 29 2019

Magma

F:=Factorial; B:=Binomial; [ (&+[(&+[(&+[F(n)/(B(n,k)*B(k,m)*B(m,j)): j in [0..m]]): m in [0..k]]): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 29 2019

Maple

seq( (n+3)!*add(add(add( Beta(k+3,n-k+1)*Beta(m+2,k-m+1)*Beta(j+1,m-j+1), j=0..m), m=0..k), k=0..n), n=0..20); # G. C. Greubel, Dec 29 2019

Mathematica

Table[(n+3)!*Sum[Beta[k+3, n-k+1]*Beta[m+2, k-m+1]*Beta[j+1, m-j+1], {k,0,n}, {m,0,k}, {j,0,m}], {n,0,20}] (* G. C. Greubel, Dec 29 2019 *)

PARI

vector(21, n, my(b=binomial); sum(k=0,n-1, sum(m=0,k, sum(j=0,m, (n-1)!/(b(n-1,k)*b(k,m)*b(m,j)) )))) \\ G. C. Greubel, Dec 29 2019

Sage

b=binomial; [sum(sum(sum(factorial(n)/(b(n,k)*b(k,m)*b(m,j)) for j in (0..m)) for m in (0..k)) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Dec 29 2019

Formula

a(n) = Sum_{k=0..n} A090595(k)*(n-k)!.

a(n) = Sum_{a+b+c+d = n} a!*b!*c!*d!.

a(n) = Sum_{k=0..n} A003149(k)*A003149(n-k).

G.f.: (Sum_{k>=0} k!*x^k)^4.

From G. C. Greubel, Dec 29 2019: (Start)

a(n) = (n+3)!*Sum_{k=0..n} Sum_{m=0..k} Sum_{j=0..m} Beta(k+3, n-k+1)*Beta(m+2, k-m+1)*Beta(j+1, m-j+1), where Beta(x,y) is the Beta function.

a(n) = Sum_{k=0..n} Sum_{m=0..k} Sum_{j=0..m} n!/(binomial(n,k) * binomial(k,m) * binomial(m,j)). (End)